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After going through the following question on Penrose Tiling and reading de Bruijn's papers on the subject, I came accross Grünbaum and Shephardbook "Tilings and Patterns", p. 543, where they say that there are only two Penrose tilings with global 5-rotational symmetry. Where can one find a proof for that (that these are the only ones)?

Moreover, I then saw in Wikipedia the following picture, which somehow implies that there are more than two tilings with a global 5-rotational symmetry... so what is in fact wrong here? can it be that most of the tilings in this picture are do not have a global 5-rotational symmetry and it is only an illusion? Thank you, Thomas.

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  • Hmm, well I know how to generate two rotationally symmetric Penrose tilings, though I'm not sure how I would show they are the only two. Start with a sun at the origin and apply the substitution four times. This will give you a larger patch about the origin which also has a sun at the center and in the same orientation. Here is after subbing twice (couldn't find four). If you keep doing this, you will get an expanding patch which eventually covers the entire plane, and by construction has 5-fold symmetry. The same works for the star. – Dan Rust Sep 05 '14 at 19:46
  • @Daniel Rust: thank you. I found this video with several subbing stages for the sun. so the wikipedia image is eventually an illusion? – Thomas Sep 05 '14 at 20:23
  • An illusion? Sorry I don't follow. – Dan Rust Sep 05 '14 at 20:32
  • @Daniel Rust: I meant, wikipedia presents several pictures of different Penrose tilings, all of them seems to have a 5-fold symmetry. But in fact only two of them have this kind of symmetry. – Thomas Sep 05 '14 at 21:24
  • Oh I see what you mean now, sorry. I'm not sure to be honest. The ones I defined happen to be fixed points of the substitution, but that doesn't mean there aren't 5-fold symmetric Penrose tilings which aren't fixed under the substitution. Maybe the cut and project method is a better way of looking at things. – Dan Rust Sep 05 '14 at 21:28

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The list of patches in the wikipedia link are not Penrose tilings as considered by Grumbaum & Shephard, just one of them is. The former are obtained via the "cut-and-project" method, while the latter are restricted to tilings that can be obtained via inflation/deflation procedure described in the book.

In this context, the proof of the claim you ask for relies on the process of deflation: by deflating repeatedly in a 5-fold tiling you will eventually reach one of the two vertices with 5-fold symmetry (in the case of Kite and Dart tilings, either a sun or a star).

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